Sandalwood and Mysore has an inseparable connection.
So what's sandalwood by the way?
This is a tree grown in the forest. The Western Ghat forest region around Mysore had vast reserves of this exotic variety called Indian Sandalwood (Santalum album) or Mysore Sandalwood.
The used part of the tree is its wood. This is used for making many things like for carvings, paste and most importantly for making sandalwood oil. Sandalwood has a  mild yet pleasing aroma. What is unique is the wood piece retains its aroma for many decades.

Sandalwood is a slow growing tree and it takes many decades before it is harvested (by felling the tree).
The oil is extracted from the wood. This is a hugely expensive item to buy ( may be around Rs 60,000 ($1500) a kg ).
Cosmetic industry (soaps, perfumes etc) use them as a natural fragrant. The abundance of sandalwood in Mysore in fact led to the creation of a Government Sandalwood Oil factory around 100 years back by the ruling Maharaja of Mysore, Krishna Raja Wodeyar IV. The factory still produces sandalwood based cosmetics.
The flip side: Once upon a time , the forests around Mysore boast of the over abundance of  sandalwood trees. Not anymore. The demand and high value of the tree trunk led to widespread tree felling and smuggling. The forest brigand Veerappan (a la Robin Hood legend for some) was often quoted with sandalwood smuggling around the forests of Mysore. Now much of the trees left, either in the forest or even in the private lands, are government controlled. Means no one can cut it without government permission.
If you are curious, you can see Sandalwood trees in many places  Mysore city. There are quite a few in the Kukkarahalli Lake and Karainji Lake area.
The nickname: The word Sandalwood is so closely associated with the region, the local cinema industry is often referred  as 'Sandalwood' in rhyme with the much famous Hollywood and Bollywood.
Souvenir: You can still get a piece of genuine product in Mysore.  Look for Mysore Sandal Soap. They are available as single soap packets as well as the so called gift packs. Then there is perfume with sandalwood fragrance. You can even buy small pieces of raw sandalwood or even the chips. These makes a good souvenir or gift article of your Mysore tour.
See to that you are not coned with some fake things, especially when it comes to the expensive sandal oil. Many things that is claimed as sandal products might not be so, if it looks like a bargain.  If you are not sure, buy things only from the state owned Cauvery Emporium (there are only two in the city, one is on Sayyaji Rao road and the other is inside Mysore Palace). They have some sections specifically for sandalwood products.